monty hall

THE MONTY HALL

WELCOME My name is Ercan Cem

I will try to present

the most famous* puzzle of all times.

the most famous* puzzle of all times.

*arguable

It is known as The Monty Hall

POPULAR BECAUSE

COUNTER INTUITIVE

THE PUZZLE

You are in a TV show.

There are three doors.

1 2 3

Behind one of them there's a prize.

It is yours if you guess correctly.

The format of the show is...

First, you make a guess.

1 2 3

The host opens an empty door from the other two.

1 2 3

The host knows where the prize is.

He offers you a new option:

You can stick with your initial choice.

OR

Select the other (unopened) door.

QUESTION

WHAT IS THE RATIONAL

STRATEGY?

IS IT...

Stick?

1 2 3

OR...

Select the other?

1 2 3

OR...

Is this just a random guess?

1 2 3

The usual* answer is that the final choice is just a random guess.

The usual* answer is that the final choice is just a random guess.

*underestimate

People tend to think that

at first, each door has the same one in three chance.

(which is true)

So, the chance that our guess is correct

is one in three.

(which is true)

Once one of the doors is opened

1 2 3

the unopened two doors

1 2 3

share the chance of the opened door.

THEREFORE

The final decision is a random guess.

1 2 3

WRONG!

The answer is:

If you want to increase your chance,

you must switch to the other door.

1 2 3

(we exclude the cases)

(that you are superstitious about your inital guess)

OR

You are a clairvoyant.

HERE IS THE EXPLANATION

(without getting too technical)

FIRST TRY

When you made your initial guess,

1 2 3

your chance of being correct was

one in three.

ALSO,

As a fact, we know that

at least one of the remaining

doors is empty.

AGREE?

GOOD.

FURTHERMORE,

The host will always have a choice to open

an empty door.

SO,

when he opens an empty door,

1 2 3

he does not provide an extra information.

He does not provide anything new.

We already know that at least one

other door is empty.

The chance that our initial guess is correct

is one in three.

Nothing changed since then.

After the door is opened,

that chance is still one in three.

HOWEVER

When two doors are left,

it cannot be the case that each has a chance

of one in three.

The TOTAL chance must add up to 1.

THEREFORE

The chance that the prize is behind the other

door is two in three.

UNLESS

You are very supertitious,

OR

CLAIRVOYANT

You must switch.

Not convinced?

SECOND TRY

Imagine that just before the host opens one of the remaining doors,

the aliens kidnap him.

Right at this moment, we can reason as follows:

The host would either open Door- 2, or Door -3.

Say he opened Door -2

1 2 3

Then, only Door -1 and Door-3 would be left.

1 2 3

The chance that the prize is behind Door-1

would be 50%.

Instead, say he opened Door -3

1 2 3

Then, only Door -1 and Door-2 would be left.

1 2 3

The chance that the prize is behind Door-1

would be 50%.

Hmmm!

Isn't that reasoning a bit...

NAIVE?

BECAUSE

It suggests that

the chance that our initial guess is correct was 50% in the first place.

We KNOW that that is WRONG!

It is one in three.

It cannot increase all of a sudden.

THEREFORE

The chance that the prize is behind the other

door is two in three.

Not convinced?

LAST TRY

LAST TRY (You’d better be convinced this time.)

Imagine a deck of cards.

(52 cards that is.)

You pick one.

If it is the ace of spades,

you win.

You picked one.

51 cards are left.

I KNOW which card is the ace of spades.

Among the other 51 cards,

at least 50 of them is NOT the ace of spades.

I turn 50 cards upside down.

Right now, there are two unopened cards on the table...

yours, and the 52nd card.

Do you REALLY

think that

the chance that your pick is the ace of spades is 50%?

TIME TO WRAP UP

In the last example, the essence of the

puzzle is much clearer.

That is because we were dealing with larger quantities.

Things look fuzzier when dealing with only three objects.

SIDE NOTE

In the original problem, suppose that the host opens a random door.

(Translation: It could be the case that the opened door is the one with the prize.)

If the opened door is empty,

then the chance that our initial guess is correct indeed rises to one in two.

HENCE

Everything depends on

whether the door was opened at random,

or with the knowledge that where the prize is.

Thanks for sparing your time.

(A simple Google search will get you to endless sources on Monty Hall. There’s even a book devoted on it.)

(If you enjoyed this puzzle, you can find more at my blog Mathzzle: puzzle.ercancem.com.)

UNTIL NEXT TIME

SEE YA!